Five for Five

I just completed this without having to change the colour of the cart. What you need to do is capture the cart only, i.e. be there when it gets inside the depot. Of course, it has to be in your team's possession.
My advice is to just stay away from the middle one, because that's where all the action is, and stick to the left or right, according to the situation. If you have to pass through the middle, stick to the wall so as to avoid enemies.
Pay attention that if your team catches many other carts aside from your own, the time won't be enough for you to complete the achievement. Each cart capture gives ~ 100 points and cart possession gives ~ 1point per 2sek.
Bringing a friend (healer!) is a good idea, but it can be done solo in a random battleground too.

/bump for Taoma

As at 20 Feb 2013 all you have to do is to "be there when it gets inside the depot" as Taoma claims.

You need not need to be there when it changes colour to your faction.

I have finally cracked the code on this one. I just got it, and the key is in the term "capture" **. You need to be there when it changes colour to your faction AND be there when it DELIVERS at the end. Repeat 5 times. Without dying.

** It's actually the Escort Service that is wrong, in their usage of the word capture. That achiev only requires you to cahnge the color of the cart, while all the other achievements referring to "capture" means to ALSO be there at the delivery.

I have been trying on this for a few times now, and I'm pretty sure you need to NOT die at all. In one round i died very early, before i could cap antyhing. Then i stayed alive for a long time and got 6 or 7 caps (note, not deliveries, but cart caps) and nothing.